Detecting Intrinsic Global Geometry of an Obstacle via Layered Scattering

Leonid Bunimovich, Gabriel Katz

correcthigh confidence

Category: Not specified
Journal tier: Specialist/Solid
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves (conditionally on Hypothesis A: zero Liouville volume of the trapped set in SN_ε) that integrals of the billiard travel-time function over the incoming boundary determine the ε-tube volume and hence the Weyl tube invariants via the polynomial in ε^2; the candidate solution reproduces the same chain: a Santaló-type identity on ∂M, Liouville-volume factorization, Weyl’s tube expansion, and a Vandermonde recovery from finitely many layers. Minor nuances (boundary-trap set measure zero is from Lax–Phillips, degree count notation) do not alter correctness. Overall, the approaches coincide in substance.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The submission cleanly links layered scattering data to intrinsic tube invariants using established tools (Santaló-type identities and Weyl’s formula). The main theorem is conditional on a natural measure-zero trapping hypothesis; the authors mitigate this with dispersing approximations. Clarity would benefit from explicitly distinguishing boundary-trapped versus interior-trapped sets, standardizing constants, and clarifying degree counts. These are modest editorial issues rather than conceptual flaws.